A single$-$bodied helical$-$coil torsion spring has a wire diameter of $1\mathrm{.}5$ mm and an outside coil diameter of $12$ mm $.$ The tightly

coiled spring has $9\mathrm{.}5$ coils, with an end extension of $l=10mm$ from the coil center to the point of load application at each end.

The spring material is a steel alloy with $S_{ut}=2030$MPa. Let $E=200$GPa and $S_y=0\mathrm{.}78S_{ut}$

a$.$ What is the maximum operating torque that can be applied for static loading?

b$.$ Estimate the coil diameter when the spring is subjected to the torque from part $($a$).$

c$.$ If the applied moment fluctuates between $0\mathrm{.}1$ Nm and $0\mathrm{.}25$ Nm $,$ determine the factor of safety against fatigue. Use a fatigue life of one

million cycles, and non$-$shot peened ASTM A$228$ steel for fatigue date. The Gerber fatigue criterion is to be used for this part, with

endurance limits defined from a repeated bending test.